k-Adaptive Routing for the Robust Network Loading Problem

TL;DR: The k-adaptive routing scheme, based on the fact that the decision-maker chooses k second-stage solutions and then commits to one of them only after realization of the uncertainty, is experimented for the Robust Network Loading problem.

About: This article is published in Electronic Notes in Discrete Mathematics.The article was published on 2018-02-01 and is currently open access. It has received 3 citations till now. The article focuses on the topics: Routing (electronic design automation) & Iterative method.

Summary

1 Introduction

• This study is about a robust linear optimization approach for the network loading problem under demand uncertainty (RNL).
• An alternative routing scheme that can be considered, studied in [2], is the kadaptive routing, where the decision-maker chooses k second-stage solutions, and then commits to one of them only after seeing the realization of the uncertainty.
• Hence, a goal when using k-adaptability is to identify a partitioning scheme that is near the efficient frontier of this trade-off.
• The information above is used to construct partitions of the uncertainty set, leading to a k-adaptable formulation of the problem with potentially improved objective function value.
• Each step is further explained in the solution strategy below.

2 Problem definition

• The authors work with two cases: one in which flows are unsplittable, or nonbifurcated, and must use a single path, and another one in which flows are splittable, or bifurcated, and can be fractionally split along several paths.
• Each commodity q ∈ Q is associated with the uncertain demand dq, within a given uncertainty set.
• In many practical applications there are limits on the number of different configurations of paths that can be implemented, so that the authors exercise the flexibility of routing schemes to mitigate the static solution conservativeness even with this simplification.
• They were predetermined as many shortest paths weighted by edges costs for each commodity.

3 Solution strategy

• In [3], the authors identify that there is a set of active uncertain parameters ξ̂ that restricts the objective function.
• This shows that the authors must partition the uncertainty set in such a way as to guarantee that the uncertain parameters for the active constraints do not all lie in one set of the partition.
• They impose that each element selected of Ξ̂k belongs to a single set of the new partition.
• Benders Decomposition Dynamic partitioning suffers from an inherent characteristic of multiplying the number of variables and constraints after each iteration.
• Specifically, given an integer solution x̃ij for the master problem (typically obtained at an integer node of the branch-and-bound tree of the master problem), the authors solve a feasibility subproblem for each element of the partition k ∈ K.

4 Implementation and Results

• The purpose of their experiment is two-fold: For nonbifurcated flows, the authors show the cost reductions provided by k-adaptive routing scheme over static.
• For bifurcated flows, the authors compare the solution times and costs of k-adaptive routing scheme with those of static and volume routings.
• The authors run all algorithms according to configuration Table 1c.
• For Decomposition method, Benders primal subproblems were solved through robust deterministic reformulation set using JuMP lazy callback functions.
• The cost reduction is higher when the protection level is higher (25%).

5 Conclusions

• The k-adaptive routing scheme was able to provide improved solutions when compared to affine and static solutions.
• The method utilized does suffer of dimensionality issues so that special techniques to maintain tractability are fundamental.
• In fact, the k-adaptive partial partitioning can provide good results, when compared to full partitioning and have better time performance.
• The authors preliminary results also showed that Benders decomposition can be efficient to speed up instances.

Figures

Table 3 Bifurcated Flows Time Performance (s)

Table 2 Bifurcated (BF) and Non Bifurcated (NBF) Flows Solution

Table 1 Problem Set Up

Frequently asked questions

Q1. What are the contributions in "K -adaptive routing for the robust network loading problem" ?

The method has an inherent characteristic of multiplying the number of variables and constraints after each iteration, so that additional measures are introduced in the solution strategy in order to control its tractability. 

Q2. What is the main characteristic of dynamic partitioning?

Benders Decomposition Dynamic partitioning suffers from an inherent characteristic of multiplying the number of variables and constraints after each iteration. 

Q3. What is the definition of k-adaptive routing?

Full partitioning means k-adaptive routing scheme where at the end of each iteration the authors add active uncertain parameters for all capacity constraints (1) of their formulation. 

Q4. What is the definition of partial routing?

Partial partitioning means k-adaptive routing scheme where (i) the authors only add active uncertain parameters referent to the set of the partition that is restricting the objective value and (ii) the authors restrict to 10 active uncertain parameters referent to the constraints of this set with minimum slack. 

Q5. What is the way to solve the subproblem?

The subproblem are solved through the the dual simplex method to leverage the fact that only the right hand side (x̃ij) of constraints change. 

Q6. How many iterations of the volume routing scheme were used?

The volume routing scheme, a variant of affine routing and only valid for bifurcated flows, was implemented based on [8], where each path variable was defined as ykqp = y 0k qp + y 1k qpdq. 

Q7. What is the nested approach to partitioning?

their approach creates a partition tree, called nested partitioning, where the children of an element represent the sets partitioning the element. 

Q8. What is the goal of the authors' experiment?

In [3] and [15] the authors independently introduce strategies to re-optimize the solution based on a new k-partitioning of the uncertainty set. 

Q9. What is the purpose of this experiment?

This is the first attempt to improve static solutions as the integrality of the second stage variables prevents us from enumerating the extreme points of Ξ (see [14]) or using classical decomposition algorithms (see [7]).• 

Q10. What is the optimal solution cost of subproblem k?

If the optimal solution cost of subproblem k is positive, the authors use the dual optimal solution (π̃ij, µ̃q) to add a strengthened Benders cut to the master problem:∑ ij∈E d π̃ij m exij ≤ d ∑ q∈Q−µ̃q m e,where m = min ij∈E π̃ij. 

Q11. What is the objective function of the problem?

The information above is used to construct partitions of the uncertainty set, leading to a k-adaptable formulation of the problem with potentially improved objective function value.• 

Q12. How is the affine adaptability used in the RNL?

This difficulty has been addressed in the literature (see [11] and [14]) by restricting the second stage variables to be affine functions of the uncertain data, in a routing scheme called affine adaptability that provides intermediary solutions between static and dynamic. 

Q13. What is the simplest way to partition the uncertainty set?

This shows that the authors must partition the uncertainty set in such a way as to guarantee that the uncertain parameters for the active constraints do not all lie in one set of the partition.